I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand the 2-torsion part better.

Is there a reference which describes the difference between $p(E\oplus F)$ and $p(E)\cdot p(F)$, perhaps in terms of Bocksteins of Stiefel-Whitney classes of $E$ and $F$?

This question was previously part of Whitney sum formula for Pontryagin classes I; Qiaochu Yuan's answer to that question might be helpful.


1 Answer 1


Brown, Edgar H., Jr. The cohomology of BSOn and BOn with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288.

Theorem 1.6, last sentence:

Under Whitney sum, $p_q\mapsto \sum_j r_{2q-j}\otimes r_j$, where $r_{2s} = p_s$ and $r_{2s+1} = (\delta w_{2s})^2+ p_s\delta w_1$.

  • 2
    $\begingroup$ I don't suppose anyone wants to provide that sentence here to have the question and answer in one place? $\endgroup$ Nov 5, 2015 at 7:08
  • $\begingroup$ @GregFriedman: I do. $\endgroup$
    – user78588
    Nov 5, 2015 at 8:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.